Question-13
The sum of the elements in each row of \(A\in \mathbb{R}^{n\times n}\) is \(1\). If \(B=A^{3} -2A^{2} +A\), which one of the following statements is correct (for \(x\in \mathbb{R}^{n}\))?
We can factor \(B\) as:
\[ \begin{aligned} B & =A^{3} -2A^{2} +A\\ & =A\left( A^{2} -2A+I\right)\\ & =A( A-I)^{2} \end{aligned} \]
Now, if the sum of elements in each row of \(A\) is \(1\), then the sum of the columns of \(A\) is equal to the vector \(v=( 1,\cdots ,1)\). From here, we see that \(Av=v\). That is, \(v\) is an eigenvector of \(A\) with eigenvalue \(1\). It follows that \(v\) is also an eigenvector of \(A-I\) with eigenvalue \(0\). If \(( A-I) v=0\), then \(( A-I)( kv) =0\) for \(k\in \mathbb{R}\). Therefore, the system \(( A-I) x=0\) has infinitely many solutions. It follows that \(Bx=0\) has infinitely many solutions.